Abstract: The present paper has two goals. First to present a natural example of a new
class of random fields which are the variable neighborhood random fields. The
example we consider is a partially observed nearest neighbor binary Markov
random field. The second goal is to establish sufficient conditions ensuring
that the variable neighborhoods are almost surely finite. We discuss the
relationship between the almost sure finiteness of the interaction
neighborhoods and the presence/absence of phase transition of the underlying
Markov random field. In the case where the underlying random field has no phase
transition we show that the finiteness of neighborhoods depends on a specific
relation between the noise level and the minimum values of the one-point
specification of the Markov random field. The case in which there is phase
transition is addressed in the frame of the ferromagnetic Ising model. We prove
that the existence of infinite interaction neighborhoods depends on the phase.